Question

A right triangle with coordinates A (－1, 4), B(－1, 8) and C (3, 4) is first rotated 90 degrees counterclockwise and then translated 2 units left and 3 units down to form Triangle A’B’C’.

What is the measure of angle B’A’C’, in degrees, in the resulting figure?

Answer 1

Answer:

The ∠B prime A prime C prime  in the resulting figure = 90°

Step-by-step explanation:

The coordinates of the right triangle = A( -1, 4), B(-1, 8) and C(3, 4)

The rotation 90° counterclockwise of a point on a preimage (x, y) gives the coordinates of the location of the point of the image after rotation as (-y, x)

Therefore we have;

Applying 90° counterclockwise rotation we have;

• The point A(-1, 4) is relocated to the point (-4, -1)
• The point B(-1, 8) is relocated to the point (-8, -1)
• The point C(3, 4) is relocated to the point (-4, 3)

Applying a translation T(-2, -3) to the new points above which is a translation 2 units left and 3 units down, we have;

The coordinates of the point A prime is (-4 - 2, -1 - 3) = (-6, -4)

The coordinates of the point B prime is (-8 - 2, -1 - 3) = (-10, -4)

The coordinates of the point C prime is (-4 - 2, 3 - 3) = (-6, 0)

The coordinates of the vertices of the triangle ΔA prime B prime C prime are A prime (-6, -4) B prime (-10, -4) C prime (-6, 0)

The measure of the angle ∠B prime A prime C prime  is given as follows;

Length of a segment of each segment of the triangle are found using the following equation;

$l = \sqrt{\left (x_{2}-x_{1} \right )^{2}+\left (y_{2}-y_{1} \right )^{2}}$

Which gives;

Length of B prime A prime = √(((-10) - (-6))² + ((-4) - (-4))²) = 4

Length of B prime C prime = √(((-10) - (-6))² + ((-4) - 0)²) = 32 = 4·√2

Length of C prime A prime = √(((-6) - (-6))² + ((0) - (-4))²) = 4

∴ B prime C prime is the hypotenuse side and the ∠B prime A prime C prime = The angle opposite to the hypotenuse side = 90°

The ∠B prime A prime C prime  in the resulting figure = 90°